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Kronecker :: rationalNormalForm

rationalNormalForm -- rational normal form of a matrix

Synopsis

Description

This function produces a matrix B in rational normal form, and invertible matrices P and Q such that P*Q = I and B = P*A*Q.
i1 : R = ZZ/101[x]

o1 = R

o1 : PolynomialRing
i2 : M = R^4

      4
o2 = R

o2 : R-module, free
i3 : A = random(M,M)

o3 = | 50  38 -40 -40 |
     | 10  27 -6  14  |
     | -36 47 32  -37 |
     | -49 21 37  -16 |

             4       4
o3 : Matrix R  <--- R
i4 : factor det(x*id_M - A)

               3      2
o4 = (x + 44)(x  - 36x  + 19x - 33)

o4 : Expression of class Product
i5 : (B,P,Q) = rationalNormalForm A

o5 = (| 1 0   0 0 |, | 18 21  20  20  |, | 50 -17 47  -18 |)
      | 0 36  1 0 |  | 5  -11 -34 -50 |  | -2 -37 49  1   |
      | 0 -19 0 1 |  | -2 -36 -30 -31 |  | 16 45  -12 0   |
      | 0 33  0 0 |  | 12 15  -33 30  |  | 39 -6  -6  0   |

o5 : Sequence
i6 : B - P*A*Q == 0

o6 = true
i7 : P*Q - id_M == 0

o7 = false

Ways to use rationalNormalForm :